- Hearing music in different environments
- Acoustics of enclosed spaces
- The direct sound
- Early Reflections
- The effect of absorption on early reflections
- Reverberant sound
- The behaviour of the reverberant sound field
- The balance of reverberant to direct sound
- The level of the reverberant sound in the steady state
- Calculating the critical distance
- The effect of source directivity on the reverberant sound
- Brain Bullets

In this chapter we will examine the behaviour of the sound in a room with particular reference to how the room's characteristics affect the quality of the perceived sound. We will also examine strategies for analysing and improving the acoustic quality of a room. Finally we will look at how we hear sound direction and consider how this affects the design of listening rooms, studios and control rooms to give good stereo listening environment.

In Chapter 1 the concept of a wave propagating without considering any boundaries was discussed. However most music is listened to within a room, and is therefore influenced by the presence of boundaries, and so it is important to understand how sound propagates in such an enclosed space. Figure 6.1 shows an idealised room with a starting pistol and a listener; assume that at some time (t = 0) that the gun is fired. There are three main aspects to how the sound of a gun behaves in the room which are as follows.

After a short delay the listener in the space will hear the sound of the starting pistol, which will have travelled the shortest distance between it and the listener. The delay will be a function of the distance, as sound travels 344 metres (1129 feet) per second or approximately 1 foot per millisecond. The shortest path between the starting pistol and the listener is the direct path and therefore this is the first thing the listener hears.

**Fig 6.1 An idealised room with an impulse from a pistol**

This component of the sound is called the direct sound and its propagation path and its associated time response is shown in Figure 6.2. The direct component is important because it carries the information in the signal in an uncontaminated form. Therefore a high level of direct sound is required for a clear sound and good intelligibility of speech.

**Fig 6.2 The direct sound in a room **

The direct sound also behaves in the same way as sound in free space, because it has not yet interacted with any boundaries. This means that we can use the equation for the intensity of a free space wave some distance from the source to calculate the intensity of the direct sound. The intensity of the direct sound is therefore given, from Chapter 1, by:

I_{direct sound} = (QW_{source})/(4πr^{2})

(6.1)

where

- I direct sound = the sound intensity (in W m
^{-2}) - Q = the directivity of the source (compared to a sphere)
- W
_{source}= the power of the source (in W) - and r = the distance from the source (in m)

Equation 6.1 shows that the intensity of the direct sound reduces as the square of the distance from the source, in the same way as a sound in free space. This has important consequences for listening to sound in real spaces. Let us calculate the sound intensity of the direct sound from a loudspeaker.

Example 6.1 A loudspeaker radiates a sound intensity level of 102 dB at 1m. What is the sound intensity level (I direct) of the direct sound at a distance of 4 m from the loudspeaker?

The sound intensity of the direct sound at a given distance can be calculated, using Equation 1.18 from Chapter 1, as:

IL = 10 log_{10}
(
W_{source}/W_{reference})
- 20 log_{10}(r) - 11 dB

As we already know the intensity level at 1 m this equation becomes:

I direct sound = I 1m - 20 log_{10}(r)

which can be used to calculate the direct sound intensity as:

I direct sound = 102 dB - 20 log_{10}(4) = 102 dB - 12 dB = 90 dB

Example 6.1 shows that the effect of distance on the direct sound intensity can be quite severe.

A little time later the listener will then hear sounds which have been reflected off one or more surfaces (walls, floor, etc.), as

**Fig 6.3 The early reflections of a room **

shown in Figure 6.3. These sounds are called early reflections and they are separated in both time and direction from the direct sound. These sounds will vary as the source or the listener move within the space. We use these changes to give us information about both the size of the space and the position of the source in the space. If any of these reflections are very delayed, total path length difference longer than about 30 milliseconds (33 feet), then they will be perceived as echoes. Early reflections can cause interference effects, as discussed in Chapter 1, and these can both reduce the intelligibility of speech, and cause unwanted timbre changes in music, in the space. The intensity levels of the early reflections are affected by both the distance and the surface from which they are reflected. In general most surfaces absorb some of the sound energy and so the reflection is weakened by the reflection process. However it is possible to have surfaces which 'focus' the sound, as shown in Figure 6.4, and in these circumstances the intensity level at the listener will be enhanced. It is important to note, however, that the total power in the sound will have been reduced by the interaction with the surface. This means that there will be less sound intensity at other positions in the room. Also any focusing structure must be large when

**Fig 6.4 A focusing surface **

measured with respect to the sound wavelength, which tends to mean that these effects are more likely to happen for high-, rather than low-frequency components. In general therefore the level of direct reflections will be less than that predicted by the inverse square law due to surface absorption. Let us calculate the amplitude of an early reflection from a loudspeaker.

Example 6.2 A loudspeaker radiates a peak sound intensity of 102 dB at 1 m. What is the sound intensity level (I reflection), and delay relative to the direct sound, of an early reflection when the speaker is 1.5 m away from a hard reflecting wall and the listener is at a distance of 4 m in front of the loudspeaker?

The geometry of this arrangement is shown in Figure 6.5 and we can calculate the extra path length due to the reflection by considering the 'image' of the loudspeaker, also shown in Figure 6.6, and by using Pythagoras' theorem. This gives the path length as 5 m.

Given the intensity level at 1 m, the intensity of the early reflection can be calculated because the reflected wave wiIl also suffer from an inverse square law reduction in amplitude:

I early reflection = I 1m - 20 log_{10} (Path length) (6.2)

Which can be used to calculate the direct sound intensity as:

I early reflection = 102 dB - 20 log_{10}(5) = 102 dB - 14 dB = 88 dB

Comparing this with the earlier example we can see that the early reflection is 2 dB lower in intensity compared to the direct sound. The delay is simply calculated from the path length as:

Delay early reflection = Path length/344 ms-1 = 5 m/344ms-1 = 14.5 ms

Similarly the delay of the direct sound is:

**Fig 6.5 A geometry for calculating the intensity of an early reflection **

**Fig 6.6 The maximum bounds for early reflection assuming no absorption or focusing **

Delay direct = r/344 ms-1 = 4m/344 ms-1 = 11.6 ms

So the early reflection arrives at the listener 14.5 ms - 11.6 ms = 2.9 ms after the direct sound. Because there is a direct correspondence between delay, distance from the source and the reduction in intensity due to the inverse square law, we can plot all this on a common graph (,ee Figure 6.6), which shows the maximum bounds of the intensity level of reflections, providing there are no focusing effects.

How does the absorption of sound affect the level of early reflections heard by the listener? The absorption coefficient of a material defines the amount of energy, or power that is removed from the sound when it strikes it. In general the absorption coefficient of real materials will vary with frequency but for the moment we shall assume they do not. The amount of energy, or power removed by a given area of absorbing material will depend on the energy, or power, per unit area striking it. As the sound intensity is a measure of the power per unit area this means that the intensity of the sound reflected is reduced in proportion to the absorption coefficient That is

Intensity reflected = Intensity incident x (1 - alpha)

where

- Intensity reflected = the sound intensity reflected after absorption (in W m-2)
- Intensity incident = the sound intensity before absorption (in W m-2)
- alpha = the absorption coefficient

Because a multiplication of sound levels is equivalent to adding the decibels together as shown in Chapter 1, Equation 6.3 can be expressed directly in terms of the decibels as

I absorbed = I incident +10log(1-alpha) (6.4)

which can be combined with Equation 6.2 to give a means of calculating the intensity of an early reflection from an absorbing surface:

I early reflection = I 1m -20log_{10}(Path length) - 10 log(1-alpha) (6.5)

As an example consider the effect of an absorbing surface on the level of the early reflection level calculated earlier.

Example 6.3 A loudspeaker radiates a peak sound intensity of 102 dB at 1 m. What is the sound intensity level (I early reflection) of an early reflection, when the speaker is 1.5 m away from a reflecting wall and the listener is at a distance of 4 m in front of the loudspeaker, and the wall has an absorption of 0.9,0.69, 0.5?

As we already know the intensity level at 1m the intensity of the early reflection can be calculated using Equation 6.5 because the reflected wave also suffers from an inverse square law reduction in amplitude:

Iearly reflection = I 1m - 20 log_{10}(Path length) + 10 log(1 - alpha)

The path length, from the earlier calculation, is 5 m so the sound intensity at the listener for the three different absorption coefficients is:

I early reflection (alpha = 0.9) = 102 dB - 20 log_{10}(5 m) + 10 log( 1- 0.9) = 102 dB - 14 dB - 10 dB = 78 dB

I early reflection (alpha = 0.69) = 88 dB + 10 log(1 - 0.69) = 88 dB - 5 dB = 83 dB

I early reflection (alpha = 0.5) = 88 dB + 10 logO - 0.5) = 88 dB - 3 dB = 85 dB

At an even later time the sound has been reflected many times and is arriving at the listener from all directions, as shown in Figure 6.7. Because there are so many possible reflection paths, each individual reflection is very close in time to its neighbours and thus there is a dense set of reflections arriving at the listener. This part of the sound is called reverberation and is desirable as it adds richness to, and supports, musical sounds. Reverberation also helps integrate all the sounds from an instrument so that a listener hears a sound which incorporates all the instrument's sounds, including the directional parts. In fact we find rooms which have very little reverberation uncomfortable and generally do not like performing music in them; it is much more fun to sing in the bathroom compared to the living room. The time taken for reverberation to occur is a function of the size of the room and will be shorter for smaller rooms, due to the shorter time between reflections. In fact the time gap between the direct sound and reverberation is an important cue to the size of the space that the music is being performed in. Because some of the sound is

**Fig 3.7 The reverberant sound in a room **

absorbed at each reflection it dies away eventually. The time that it takes for the sound to die away is called the reverberation time and is dependent on both the size of the space and the amount of sound absorbed at each reflection. In fact there are three aspects of the reverberant field that the space affects, see Figure 6.8.

**The increase of the reverberant field level:**This is the initial portion of the reverberant field and is affected by the room size, which affects the time between reflections and therefore the time it takes the reverberant field to build up. The amount of absorption in the room also affects the time that it takes the sound to get to its steady state level. This is because, as shall be shown later, the steady state level is inversely proportional to the amount of absorption in the room. The sound level will take longer to reach a louder level than a smaller one, because the rate at which sound builds up depends on the time between reflections and the absorption.**The steady state level of the reverberant field:**If a steady tone, such as an organ note, is played in the space then after a period of time the reverberant sound will reach a constant level because at that point the sound power input balances the power lost by absorption in the space. This means that the steady state level will be louder in rooms which have a small amount of absorption. Note that a transient sound in the space will not reach a steady state level.- The decay of the reverberant field level: When a tone in the space stops, or after a transient, the reverberant sound level will not reduce immediately but will instead decay at a rate determined by the amount of sound energy that is absorbed at each reflection. Thus in spaces with a small amount of absorption the reverberant field will take longer to decay.

**Fig 6.8 The time and amplitude evolution of the reverberant sound in a room **

Bigger spaces tend to have longer reverberation times and well furnished spaces tend to have shorter reverberation times. Reverberation time can vary from about 0.2 of a second for a small well furnished living room to about 10 seconds for a large glass and stone cathedral.

The reverberation part of the sound in a room behaves differently, compared to the direct sound and early reflections from the perspective of the listener. The direct sound and early reflections follow the inverse square law, with the addition of absorption effects in the case of early reflections, and so their amplitude varies with position. However the reverberant part of the sound remains constant with the position of the listener in the room.

**Fig 6.9 The source of the steady state sound level of the reverberant field **

This is not due to the sound waves behaving differently from normal waves; instead it is due to the fact that the reverberant sound waves arrive at the listener from all directions. The result is that at any point in the room there are a large number of sound waves whose intensities are being added together. These sound waves have many different arrival times, directions and amplitudes because the sound waves are reflected back into the room, and so shuttle forwards, backwards and sideways around the room as they decay. The steady state sound level, at a given point in the room, therefore is an integrated sum of all the sound intensities in the reverberant part of the sound, as shown in Figure 6.9. Because of this behaviour the reverberant part of the sound in a room is often referred to as the reverberant field.

This behaviour of the reverberant field has two consequences. Firstly the balance between the direct and reverberant sounds will alter depending on the position of the listener relative to the source. This is due to the fact that the level of the reverberant field is independent of the position of the listener with respect to the source, whereas the direct sound level is dependent on the distance between the listener and the sound source. These effects are summarised in Figure 6.10 which shows the relative levels of direct to reverberant field as a function of distance from the source. This figure shows that there is a distance from the source at which the reverberant field will begin to dominate the direct field from the source. The transition occurs when the two are equal and this point is known as the critical distance.

**Fig 6.10 The composite effect of direct sound and reverberant field on the sound intensity as a function of the distance from the source **

Secondly because, in the steady state, the reverberant sound at any time instant is the sum of all the energy in the reverberation tail the overall sound level is increased by reverberation. The level of the reverberation will depend on how fast the sound is absorbed in the room. A low level of absorption will result in sound that stays around in the room for longer and so will give a higher level of reverberant field. In fact, if the average level of absorption coefficient for the room is given by a., the power level in the reverberation sound in a room can be calculated using the following equation:

W _{reverberant} = W_{source} 4 ((1-alpha)/S alpha) (6.6)

where

- W
_{source}= the reverberant sound power (in W) - S= the total surface area in the room (in m2)
- W
_{source}= the power of the source (in W) - alpha= the average absorption coefficient in the room

**Fig 6.11 The leaky bucket model of reverberant field intensity **

Equation 6.6 is based on the fact that, at equilibrium, the rate of energy removal from the room will equal the energy put into its reverberant sound field. As the sound is absorbed when it hits the surface, it is absorbed at a rate which is proportional to the surface area times the average absorption, or So.. This is similar to a leaky bucket being filled with water where the ultimate water level will be that at which the water runs out at the same rate as it flows in, see Figure 6.11. The amount of sound energy available for contribution to the reverberant field is also a function of the absorption because if there is a large amount of absorption then there will be less direct sound reflected off a surface to contribute to the reverberant field - remember that before the first reflection the sound is direct sound. The amount of sound energy available to contribute to the reverberant field is therefore proportional to the residual energy left after the first reflection, or (1 - alpha) because alpha is absorbed at the first surface. The combination of these two effects gives (1 - alpha) / S alpha the term in Equation 6.6. The factor of four in Equation 6.6 arises from the fact that sound is approaching the surfaces in the room from all possible directions. An interesting result from Equation 6.6 is that it appears that the level of the reverberant field depends only on the total absorbing surface area. In other words it is independent of the volume of the room. However in practice the surface area and volume are related because one encloses the other. In fact, because the surface area in a room becomes less as its volume decreases, the reverberant sound level becomes higher for a given average absorption coefficient in smaller rooms. Another way of visualising this is to realise that in a smaller room there is less volume for a given amount of sound energy to spread out in, like a pat of butter on a smaller piece of toast. Therefore the energy density, and thus the sound level, must be higher in smaller rooms.

The term (1 - alpha)/Salpha in Equation 6.6 is often inverted to give a quantity known as the room constant, R, which is given by:

R = S alpha/(1 - alpha) (6.7)

where

- R = the room constant (in m2) and
- alpha = the average absorption coefficient in the room

Using the room constant Equation 6.6 simply becomes:

W _{reverberant} = W_{source} (4/R)
(6.8)

In terms of the sound power level this can be expressed as:

SWL_{reverberant}
= 10 log_{10}(W_{source}/W_{reference})+ 10 log_{10} (4/R) (6.9)

As alpha is a number between 0 and 1 this also means that the level of the reverberant field will be greater in a room with a small surface area, compared to a larger room, for a given level of absorption coefficient. However one must be careful in taking this result to extremes. A long and very thin cylinder will have a large surface area, but Equation 6.6 will not predict the reverberation level correctly because in this case the sound will not visit all the surfaces with equal probability. This will have the effect of modifying the average absorption coefficient and so will alter the prediction of Equation 6.6. Therefore one must take note of an important assumption behind Equation 6.6 which is that the reverberant sound visits all surfaces with equal probability and from all possible directions. This is known as the diffuse field assumption. It can also be looked at as a definition of a diffuse field. In general the assumption of a diffuse field is reasonable and it is usually a design goal for most acoustics. However it is important to recognise that there are situations in which it breaks down, for example at low frequencies.

As an example consider the effect of different levels of absorption and surface area on the level of the reverberant field that might arise from the loudspeaker described earlier.

Example 6.4 A loudspeaker radiates a peak sound intensity of 102 dB at 1 m. What is the sound pressure level of the reverberant field if the surface area of the room is 75 m2, and the average absorption coefficient is (a) 0.9, and (b) 0.2? What would be the effect of doubling the surface area in the room while keeping the average absorption the same?

From Equation 1.18 we can say:

SIL = 10 log_{10}
(
W_{source}/W_{reference}
)
- 20 log_{10}(r) - 11 dB

Thus the sound power level (SWL) radiated by the loudspeaker is:

SWL = 10 loglo
(W_{source}/W_{reference})
=SIL + 11 dB = 102 dB + 11 dB = 113 dB

The power in the reverberant field is given by:

SWLreverberent = 10 log_{10}( W_{source}/W_{reference}) + 10 log_{10}(4/R)

The room constant 'R' for the two cases is:

R(alpha = 0.9) = S alpha/(1-alpha) = (75 m2 x 0.9)/(1-0.9) = 675 m2

R(alpha = 0.2) = S alpha/(1-alpha) = (75 m2 x 0.2)/(1-0.2) = 18.75 m2

The level of the reverberant field can therefore be calculated from:

SWL reverberent = 10 log_{10} (W_{source}/W_{reference})+ 10 log_{10} (4/R)
= 113 dB + 10 log_{10}(4/R)

which gives:

SWL reverberent (alpha = 0.9) = 113 dB + 10 10 log_{10}(4/675)
= 113 dB - 22.3 dB = 90.7 dB

and:

SWL reverberent (alpha = 0.2) = 113 dB + 10 10 log_{10}(4/18.75) = 113 dB - 6.7 dB = 106 dB

The effect of doubling the surface area is to increase the room constant by the same proportion, so we can say that:

SWL_{reverberant} (S_{doubled}) = 10 log(W_{source}/W_{reference})+ 10 log_{10}(4/2R)
= 113 dB + 10 log_{10} (4/R) + 10 log_{10}(1/2)

which gives:

SWL_{reverberant} (S_{doubled}) = 113 dB + 10 log_{10} (4/R) - 3 dB

Thus the effect of doubling the surface area is to reduce the level of the reverberant field by 3 dB in both cases.

Clearly the level of the reverberant field is strongly affected by the level of average absorption. The first example would be typical of an extremely 'dead' acoustic environment, as found in some studios, whereas the second is typical of an average living room. The amount of loudspeaker energy required to produce a given volume in the room is clearly much greater, about 15 dB, in the first room compared with the second. If there is a musician in the room then they will experience a 'lift' in output due to the reverberant field in the room. Because of this musicians feel uncomfortable playing in rooms with a low level of reverberant field and prefer performing in rooms which help them in producing more output. This is also one of the reasons we prefer singing in the bathroom.

The reverberant field is, in most cases, diffuse, and therefore visits all parts of the room with equal probability. Also at any point, and at any instant, we hear the total power in the reverberant field, as discussed earlier. Because of this it is possible to equate the power in the reverberant field to the sound pressure level. Thus we can say:

SPL_{reverberant} = SWL_{reverberant} = 10 log_{10}(W_{source}/W_{reference}) + 10 log_{10} (4/R) (6.10)

The distance at which the reverberant level equals the direct sound, the critical distance, can also be calculated using the above equations. At the critical distance the intensity due to the direct field and the power in the reverberant field at a given point are equal so we can equate Equation 6.1 and Equation 6.8 to give:

(QW_{source})/(4πr_{critical distance}^{2}) = W_{source}(4/R)

Which can be rearranged to give:

r_{critical distance}^{2} = (R/4)x(Q/4π)

Thus the critical distance is given by:

r_{critical distance} =((1/16π) (RQ)= 0.141(RQ)

(6.11)

Equation 6.11 shows that the critical distance is determined only by the room constant and the directivity of the sound source. Because the room constant is a function of the surface area of the room the critical distance will tend to increase with larger rooms. However many of us listen to music in our living rooms so let us calculate the critical distance for a hi-fi loudspeaker in a living room.

Example 6.5 What is the critical distance for a free standing, omnidirectional, loudspeaker radiating into a room whose surface area is 75 m2, and whose average absorption coefficient is 0.2? What would be the effect of mounting the speaker into a wall?

The speaker is omnidirectional so the 'Q' is equal to 1. The room constant 'R' is the same as was found in the earlier example, 18.75 m2. Substituting both these values into Equation 6.11 gives:

r_{critical distance} = 0.141(RQ) = 0.141(18.75 x 1 = 0.61 m (61 cm)

This is a very short distance! If the speaker is mounted in the wall the 'Q' increases to 2, because the speaker can only radiate into 2π steradians, so the critical distance increases to:

r_{critical distance} = 0.141VRQ = 0.141Y18.75 x 2 = O.86m (86cm)

Which is still quite small!

As most people would be about 2 m away from their loudspeakers when they are listening to them this means that in a normal domestic setting the reverberant field is the most dominant source of sound energy from the hi-fi, and not the direct sound. Therefore the quality of the reverberant field is an important aspect of the performance of any system which reproduces recorded music in the home. There is also an effect on speech intelligibility in the space as the direct sound is the major component of the sound which provides this. The level of the reverberant field is a function of the average absorption coefficient in the room. Most real materials, such as carpets, curtains, sofas and wood panelling have an absorption coefficient which changes with frequency. This means that the reverberant field level will also vary with frequency, in some cases quite strongly. Therefore in order to hear music, recorded or otherwise, with good fidelity, it is important to have a reverberant field which has an appropriate frequency response. As seen in the previous chapter, one of the cues for sound timbre is the spectral content of the sound which is being heard, and this means that when the reverberant field is dominant, as it is beyond the critical distance, it will determine the perceived timbre of the sound. This subject will be considered in more detail later in the chapter.

There is an additional effect on the reverberation field, and that is the directivity of the source of sound in the room. Most hi-fi loudspeakers, and musical instruments, are omnidirectional at low frequencies but are not necessarily so at higher ones. As the level of the reverberant field is a function of both the average absorption and the directivity of the source, the variation in directivity of real musical sources will also have an effect on the reverberant sound field and hence the perception of the timbre of the sound. Consider the following example of a typical domestic hifi speaker in the living room considered earlier.

Example 6.6 A hi-fi loudspeaker, with a flat-on axis, direct field, response, has a 'Q' which varies from 1 to 25, and radiates a peak on axis sound intensity of 102 dB at 1 m. The surface area of the room is 75 m2, and the average absorption coefficient is 0.2. Over what range does the sound pressure level of the reverberant field vary? As the speaker has a flat-on axis response the intensity of the direct field given by Equation 6.1 should be constant.

That is:

I_{directive source} = QW_{source}/4πr2

where

- I
_{directive source}= the sound intensity (in W m-2) - Q = the directivity of the source (compared to a sphere)
- W
_{source}= the power of the source (in W) and - r = the distance from the source (in m)

should be constant. Therefore the sound power radiated by the loudspeaker can be calculated by rearranging Equation 6.12 to give:

W_{source} = (4π/Q) I_{directive source} (6.13)

Equation 6.13 shows that in order to achieve a constant direct sound response the power radiated by the source must reduce as the 'Q' increases. The power in the reverberant field is given by:

W _{reverberant} = W_{source} (4/R)
(6.14)

By combining Equations 6.13 and 6.14 the reverberant field due to the loudspeaker can be calculated as:

W _{reverberant} = I_{directive source}(4π/Q)*(4/R)

which gives a level for the reverberant field as:

SWL_{reverberant} = 10 log_{10}
(
I directivesource/ I ref)
+ 10 log_{10} (4πr) -10 log_{10} (Q)
+ 10 log_{10} (4/R)

The room constant 'R' is 18.75 m2, as calculated in Example 6.4.

The level of the reverberant field can therefore be calculated as:

SWL_{reverberant} = 102 dB + 11 dB - 10 log_{10} (Q)
+ 10 log_{10}(4/18.75 m^{2} )

which gives:

SWL_{reverberant} (Q = 1) = 102 dB + 11 dB - 10 log_{10}(1) - 6.7 dB
= 106.3 dB

for the level of the reverberant field when the 'Q' is equal to 1, and:

SWL_{reverberant} (Q = 25) = 102 dB + 11 dB - 10 log_{10}(25) - 6.7 dB
= 92.3 dB

when the 'Q' is equal to 25.

Thus the reverberant field varies by 106.3 - 92.3 = 14 dB over the frequency range.

The effect therefore of a directive source with constant on axis response is to reduce the reverberant field as the 'Q' gets higher. The subjective effect of this would be similar to reducing the high 'Q' regions via the use of a tone control which would not normally be acceptable as a sound quality. A typical reverberant response of a typical domestic hi-fi speaker is shown in Figure 6.12. Note that the reverberant response tends to drop in both the midrange and high frequencies. This is due to the bass and treble speakers becoming more directive at the high ends of their frequency range. The dip in reverberant energy will make the speaker less 'present' and may make sounds in this region harder to hear in the mix. The drop in reverberant field at the top end will make the speaker sound 'duller'. Some manufacturers try to compensate for these effects by allowing the on-axis response to rise in these regions, however this brings other problems. The reduction in reverberant field with increasing 'Q' is used to advantage in speech systems to raise the level of direct sound above the reverberant field and so improve the intelligibility.

**Fig 6.12 The reverberant response of a domestic hi fidelity loudspeaker**

After a short delay the listener in the space will hear the sound of the starting pistol, which will have travelled the shortest distance between it and the listener. The delay will be a function of the distance, as sound travels 344 metres (1129 feet) per second or approximately 1 foot per millisecond. The shortest path between the starting pistol and the listener is the direct path and therefore this is the first thing the listener hears.

The direct component is important because it carries the information in the signal in an uncontaminated form.Therefore a high level of direct sound is required for a clear sound and good intelligibility of speech.

The direct sound also behaves in the same way as sound in free space, because it has not yet interacted with any boundaries. This means that we can use the equation for the intensity of a free space wave some distance from the source to calculate the intensity of the direct sound. The intensity of the direct sound is therefore given, from Chapter 1, by:

I_{direct sound} = (QW_{source})/(4πr^{2})

A little time later the listener will then hear sounds which have been reflected off one or more surfaces (walls, floor, etc.)

These sounds are called early reflections and they are separated in both time and direction from the direct sound. These sounds will vary as the source or the listener move within the space. We use these changes to give us information about both the size of the space and the position of the source in the space. If any of these reflections are very delayed, total path length difference longer than about 30 milliseconds (33 feet), then they will be perceived as echoes. Early reflections can cause interference effects, as discussed in Chapter 1, and these can both reduce the intelligibility of speech, and cause unwanted timbre changes in music, in the space

The intensity levels of the early reflections are affected by both the distance and the surface from which they are reflected. In general most surfaces absorb some of the sound energy and so the reflection is weakened by the reflection process. However it is possible to have surfaces which 'focus' the sound, as shown in Figure 6.4, and in these circumstances the intensity level at the listener will be enhanced. It is important to note, however, that the total power in the sound will have been reduced by the interaction with the surface. This means that there will be less sound intensity at other positions in the room

The absorption coefficient of a material defines the amount of energy, or power that is removed from the sound when it strikes it. In general the absorption coefficient of real materials will vary with frequency but for the moment we shall assume they do not. The amount of energy, or power removed by a given area of absorbing material will depend on the energy, or power, per unit area striking it. As the sound intensity is a measure of the power per unit area this means that the intensity of the sound reflected is reduced in proportion to the absorption coefficient That is

Intensity reflected = Intensity incident x (1 - alpha)

where

- Intensity reflected = the sound intensity reflected after absorption (in W m-2)
- Intensity incident = the sound intensity before absorption (in W m-2)
- alpha = the absorption coefficient

At an even later time the sound has been reflected many times and is arriving at the listener from all directions, as shown in Figure 6.7. Because there are so many possible reflection paths, each individual reflection is very close in time to its neighbours and thus there is a dense set of reflections arriving at the listener. This part of the sound is called reverberation and is desirable as it adds richness to, and supports, musical sounds. Reverberation also helps integrate all the sounds from an instrument so that a listener hears a sound which incorporates all the instrument's sounds, including the directional parts. In fact we find rooms which have very little reverberation uncomfortable and generally do not like performing music in them; it is much more fun to sing in the bathroom compared to the living room. The time taken for reverberation to occur is a function of the size of the room and will be shorter for smaller rooms, due to the shorter time between reflections. In fact the time gap between the direct sound and reverberation is an important cue to the size of the space that the music is being performed in.

The time that it takes for the sound to die away is called the reverberation time and is dependent on both the size of the space and the amount of sound absorbed at each reflection. In fact there are three aspects of the reverberant field that the space affects, see Figure 6.8.

**The increase of the reverberant field level:**This is the initial portion of the reverberant field and is affected by the room size, which affects the time between reflections and therefore the time it takes the reverberant field to build up. The amount of absorption in the room also affects the time that it takes the sound to get to its steady state level. This is because, as shall be shown later, the steady state level is inversely proportional to the amount of absorption in the room. The sound level will take longer to reach a louder level than a smaller one, because the rate at which sound builds up depends on the time between reflections and the absorption.**The steady state level of the reverberant field:**If a steady tone, such as an organ note, is played in the space then after a period of time the reverberant sound will reach a constant level because at that point the sound power input balances the power lost by absorption in the space. This means that the steady state level will be louder in rooms which have a small amount of absorption. Note that a transient sound in the space will not reach a steady state level.- The decay of the reverberant field level: When a tone in the space stops, or after a transient, the reverberant sound level will not reduce immediately but will instead decay at a rate determined by the amount of sound energy that is absorbed at each reflection. Thus in spaces with a small amount of absorption the reverberant field will take longer to decay.

Bigger spaces tend to have longer reverberation times and well furnished spaces tend to have shorter reverberation times. Reverberation time can vary from about 0.2 of a second for a small well furnished living room to about 10 seconds for a large glass and stone cathedral.

The behaviour of the reverberant sound field

The direct sound and early reflections follow the inverse square law, with the addition of absorption effects in the case of early reflections, and so their amplitude varies with position. However the reverberant part of the sound remains constant with the position of the listener in the room.This is not due to the sound waves behaving differently from normal waves; instead it is due to the fact that the reverberant sound waves arrive at the listener from all directions. The result is that at any point in the room there are a large number of sound waves whose intensities are being added together. These sound waves have many different arrival times, directions and amplitudes because the sound waves are reflected back into the room, and so shuttle forwards, backwards and sideways around the room as they decay. The steady state sound level, at a given point in the room, therefore is an integrated sum of all the sound intensities in the reverberant part of the sound, as shown in Figure 6.9. Because of this behaviour the reverberant part of the sound in a room is often referred to as the reverberant field.

The balance of reverberant to direct sound

This behaviour of the reverberant field has two consequences. Firstly the balance between the direct and reverberant sounds will alter depending on the position of the listener relative to the source. This is due to the fact that the level of the reverberant field is independent of the position of the listener with respect to the source, whereas the direct sound level is dependent on the distance between the listener and the sound source. These effects are summarised in Figure 6.10 which shows the relative levels of direct to reverberant field as a function of distance from the source. This figure shows that there is a distance from the source at which the reverberant field will begin to dominate the direct field from the source. The transition occurs when the two are equal and this point is known as the critical distance.

The level of the reverberant sound in the steady state

at equilibrium, the rate of energy removal from the room will equal the energy put into its reverberant sound field. As the sound is absorbed when it hits the surface, it is absorbed at a rate which is proportional to the surface area times the average absorption, or So.. This is similar to a leaky bucket being filled with water where the ultimate water level will be that at which the water runs out at the same rate as it flows in, see Figure 6.11. The amount of sound energy available for contribution to the reverberant field is also a function of the absorption because if there is a large amount of absorption then there will be less direct sound reflected off a surface to contribute to the reverberant field - remember that before the first reflection the sound is direct sound.

An interesting result from Equation 6.6 is that it appears that the level of the reverberant field depends only on the total absorbing surface area. In other words it is independent of the volume of the room. However in practice the surface area and volume are related because one encloses the other. In fact, because the surface area in a room becomes less as its volume decreases, the reverberant sound level becomes higher for a given average absorption coefficient in smaller rooms. Another way of visualising this is to realise that in a smaller room there is less volume for a given amount of sound energy to spread out in, like a pat of butter on a smaller piece of toast. Therefore the energy density, and thus the sound level, must be higher in smaller rooms.

Critical Distance

As most people would be about 2 m away from their loudspeakers when they are listening to them this means that in a normal domestic setting the reverberant field is the most dominant source of sound energy from the hi-fi, and not the direct sound. Therefore the quality of the reverberant field is an important aspect of the performance of any system which reproduces recorded music in the home. There is also an effect on speech intelligibility in the space as the direct sound is the major component of the sound which provides this. The level of the reverberant field is a function of the average absorption coefficient in the room. Most real materials, such as carpets, curtains, sofas and wood panelling have an absorption coefficient which changes with frequency. This means that the reverberant field level will also vary with frequency, in some cases quite strongly. Therefore in order to hear music, recorded or otherwise, with good fidelity, it is important to have a reverberant field which has an appropriate frequency response.

The effect of source directivity on the reverberant sound

There is an additional effect on the reverberation field, and that is the directivity of the source of sound in the room. Most hi-fi loudspeakers, and musical instruments, are omnidirectional at low frequencies but are not necessarily so at higher ones. As the level of the reverberant field is a function of both the average absorption and the directivity of the source, the variation in directivity of real musical sources will also have an effect on the reverberant sound field and hence the perception of the timbre of the sound.

The effect therefore of a directive source with constant on axis response is to reduce the reverberant field as the 'Q' gets higher. The subjective effect of this would be similar to reducing the high 'Q' regions via the use of a tone control which would not normally be acceptable as a sound quality. A typical reverberant response of a typical domestic hi-fi speaker is shown in Figure 6.12. Note that the reverberant response tends to drop in both the midrange and high frequencies. This is due to the bass and treble speakers becoming more directive at the high ends of their frequency range. The dip in reverberant energy will make the speaker less 'present' and may make sounds in this region harder to hear in the mix. The drop in reverberant field at the top end will make the speaker sound 'duller'. Some manufacturers try to compensate for these effects by allowing the on-axis response to rise in these regions, however this brings other problems. The reduction in reverberant field with increasing 'Q' is used to advantage in speech systems to raise the level of direct sound above the reverberant field and so improve the intelligibility.